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Cambridge University Press eBooks

The Electron Spin. Entangled States

Ana Marı́a Cetto, Luis de la Peña

Published Dec 18, 2025• 0 citations
Publisher

Abstract

Chapter 14 is entirely devoted to the electron spin and an introduction to quantum entanglement. The first part deals with the groundbreaking discovery of spin, its introduction into quantum formalism, and some of its most important effects on atomic spectra, notably the anomalous Zeeman effect. Historically, spin has been considered as an angular momentum that particles can have by the mere fact of their existence, which is called "intrinsic" and does not require any explanation. To address this shortcoming, Section 4 presents a possible explanation for the origin of electron spin as a result of its interaction with the vacuum field. Section 5 introduces the entangled system of two particles with spin, which provides an opportunity to discuss, necessarily schematically, the Schrödinger cat and the Einstein-Podolsky-Rosen thought experiment, as well as the Bell inequalities.

DOI
10.1017/9781009679633.015

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Chapter 14 is entirely devoted to the electron spin and an introduction to quantum entanglement. The first part deals with the groundbreaking discovery of spin, its introduction into quantum formalism, and some of its most important effects on atomic spectra, notably the anomalous Zeeman effect. Historically, spin has been considered as an angular momentum that particles can have by the mere fact of their existence, which is called "intrinsic" and does not require any explanation. To address this shortcoming, Section 4 presents a possible explanation for the origin of electron spin as a result of its interaction with the vacuum field. Section 5 introduces the entangled system of two particles with spin, which provides an opportunity to discuss, necessarily schematically, the Schrödinger cat and the Einstein-Podolsky-Rosen thought experiment, as well as the Bell inequalities.
First, the experimental foundations for the existence of a new dynamical variable, the spin of particles, are presented. The Pauli equation is then derived in detail, two applications are given and the energy levels of a particle with spin in a constant magnetic field are studied. This is the analysis of Landau levels, which can be performed by using the known results on the spectrum of harmonic oscillators. In this chapter we also study in detail the addition of orbital angular momentum and spin, as well as the spin–orbit interaction with the resulting Thomas precession. Stern-Gerlach experiment and electron spin The hypothesis that the electron has an intrinsic magnetic moment and an angular momentum, in short a spin , was first suggested in Uhlenbeck and Goudsmit (1926). They noticed, even before the discovery of quantum mechanics, that a complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron, and hence the concept of an electron as a point charge was insufficient. First, let us therefore try to understand how a magnetic moment can be associated to an atomic system. For this purpose, consider for simplicity the Bohr model of an hydrogen atom, where the electron moves along a circular orbit and rotates with orbital angular momentum L .
Abstract This chapter gets into the heart of the discovery of spin in physics as physicists try to make sense of the atomic spectra with the discovery of the anomalous Zeeman effect by Thomas Preston before the dawn of the twentieth century as Nature gave the first signal of the existence of quantum spin. Following Bohr, Sommerfeld, and Lande’s contributions with integers and then half-integers, quantum numbers make appearances, and after Uhlenbeck and Goudsmit’s ground-breaking contributions, physicists finally embrace the idea that electron spins. Pauli’s introduction of quantum spin with Pauli matrices provided a non-relativistic theory of the quantum spin. Hestenes argues that spin is not intrinsically related to Pauli matrices, and contrary to conventional wisdom, he advocates the view that the spin is very much present in the non-relativistic quantum physics.
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei
In the introductory remarks to Chapter 5 mention was made of the 1925 proposal of Uhlenbeck and Goudsmit that to an electron there should be assigned an intrinsic angular momentum or spin S = (S x , S y , S z )—this in addition to its quantum angular momentum (or orbital angular momentum) M = (M x , M y , M z ) defined in (5.4.4). This is not to be taken to mean that an electron actually spins about an axis in some physical way. Experiments (in particular the noted Stern-Gerlach experiment) have provided adequate justification for the proposed concept of spin, a proposal also made by R. Bichowsky and H. Urey. Dirac has shown that the notion of spin is necessary if quantum mechanics and Einstein relativity are to be consistent theories. Chemical properties of atoms, and their ability to combine and form molecules, the splitting of spectral lines by a magnetic field applied to an atom (i.e., the Zeeman effect, which we discuss later), all depend on this important notion.
This chapter discusses the prime cause of the magnetic moment within an individual atom. In the classical model, the angular momentum of the electrons can be used to determine the magnetic moments of the electrons by invoking the concept of electrical charge in motion. There are two contributions to the electronic magnetic moment: an orbital magnetic moment due to orbital angular momentum and a spin magnetic moment due to electron spin. The component of orbital angular momentum along the axis of a magnetic field is restricted to discrete values by quantum mechanics. Electrons have spin angular momentum, which can be represented by the spin quantum number. As the magnetic field is increased and the field splitting becomes greater than the multiple splitting, the anomalous Zeeman effect changes over to a normal Zeeman effect. The chapter looks at experimental evidence confirming the quantization of magnetic moments, specifically the Zeeman effect and the Stern-Gerlach experiment.
Quantum theory of electron spin is developed here based on the extended least action principle and assumptions of intrinsic angular momentum of an electron with random orientations. The novelty of the formulation is the introduction of relative entropy for the random orientations of intrinsic angular momentum when extremizing the total actions. Applying recursively this extended least action principle, we show that the quantization of electron spin is a mathematical consequence when the order of relative entropy approaches a limit. In addition, the formulation of the measurement probability when a second Stern-Gerlach apparatus is rotated relative to the first Stern-Gerlach apparatus, and the Schrödinger-Pauli equation, are recovered successfully. Furthermore, the principle allows us to provide an intuitive physical model and formulation to explain the entanglement phenomenon between two electron spins. In this model, spin entanglement is the consequence of the correlation between the random orientations of the intrinsic angular momenta of the two electrons. Since spin orientation is an intrinsic local property of the electron, the correlation of spin orientations can be preserved and manifested even when the two electrons are remotely separated. The entanglement of a spin singlet state is represented by two joint probability density functions that reflect the orientation correlation. Using these joint probability density functions, we prove that the Bell-CHSH inequality is violated in a Bell test. To test the validity of the spin-entanglement model, we propose a Bell test experiment with time delay. Such an experiment starts with a typical Bell test that confirms the violation of the Bell-CHSH inequality but adds an extra step that Bob’s measurement is delayed with a period of time after Alice’s measurement. As the time delay increases, the test results are expected to change from violating the Bell-CHSH inequality to not violating the inequality.